Question

a local café posts the number of calories in each of their full - sized sandwiches and full - sized salads. the table shows each value for number of calories and the boxplots display the calorie distributions. compare sandwiches 1110,1070,970,830,960,1080,940,810,640,760,590,720,590 salads 620,500,640,450,410,560,350,410 the stem and leaf plot shows the average points - per - game in the nba playoffs, rounded to the nearest whole point, for two different nba basketball players, michael jordan and lebron james. which of the following statements is true? michael jordan lebron james 2|3 4 9 2|5 5 6 6 7 8 8 9 2 2 1 1 1|3 0 0 1 3 4 7 6 6 5 5 5|3 5 4|4 average points per playoff game key: 4|4 = 44 a) the median of the average points - per - game for jordan is less than the median of the average points - per - game for james. b) the range of the average points - per - game for jordan is less than the range of the average points - per - game for james. c) the interquartile range is the same for both players. d) the mean of the average points - per - game for jordan is less than the median of the average points - per - game for james. e) both distributions are skewed to the right.

Explanation:

Step1: Organize Jordan's data

Jordan's data set: 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 31, 31, 31, 32, 32, 35, 35, 35, 36, 36, 37, 44. There are \(n = 22\) data - points. The median is the average of the 11th and 12th ordered values. The 11th value is 31 and the 12th value is 32, so the median of Jordan's data, \(M_J=\frac{31 + 32}{2}=31.5\).

Step2: Organize James's data

James's data set: 23, 24, 30, 30, 31, 33, 34, 35. There are \(n = 8\) data - points. The median is the average of the 4th and 5th ordered values. The 4th value is 30 and the 5th value is 31, so the median of James's data, \(M_L=\frac{30+31}{2}=30.5\).

Step3: Calculate the range for Jordan

The range of Jordan's data, \(R_J=44 - 23=21\).

Step4: Calculate the range for James

The range of James's data, \(R_L=35 - 23 = 12\). So \(R_J>R_L\).

Step5: Calculate the inter - quartile range for Jordan

First, find the first quartile \(Q_1\) and third quartile \(Q_3\) for Jordan. For \(n = 22\), the lower half has 11 data - points. \(Q_1\) is the 6th value of the lower half, which is 26. The upper half also has 11 data - points. \(Q_3\) is the 6th value of the upper half, which is 36. The inter - quartile range for Jordan, \(IQR_J=Q_3 - Q_1=36 - 26 = 10\).

Step6: Calculate the inter - quartile range for James

For \(n = 8\), the lower half is 23, 24, 30, 30 and the upper half is 31, 33, 34, 35. \(Q_1 = 24\) and \(Q_3=34\), \(IQR_L=34 - 24 = 10\).

Step7: Analyze skewness

For Jordan's data, the mean is \(\bar{x}_J=\frac{\sum_{i = 1}^{22}x_i}{22}\), \(\sum_{i=1}^{22}x_i=23 + 24+\cdots+44=667\), \(\bar{x}_J=\frac{667}{22}\approx30.32\). Since the mean (\(30.32\)) is less than the median (\(31.5\)), Jordan's data is skewed left. For James's data, the mean \(\bar{x}_L=\frac{23 + 24+30+30+31+33+34+35}{8}=\frac{240}{8}=30\), and the median is 30.5, and it is also slightly skewed left.

Answer:

C. The interquartile range is the same for both players.